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We leave the consideration of this special (easy) case and the completion of the procedure compute-BMShifts to the reader. 4. A + m] of the text is the current match; v denotes the shortest full period of the suffix of the pattern, v is a period of the current match. Shaded area is the "forbidden" part of the text. 4 * Analysis of Boyer-Moore algorithm Now we analyze the Boyer-Moore algorithm. n. n bound. The fact that the bound is linear is completely non trivial, and surprising in view of the quadratic behavior of BM algorithm when modified to search for all occurrences of the pattern.

1. KNUTH-MORRIS-PRATT ALGORITHM 23 i + j . Its maximal value is n and minimal value is 0. Each time a successful comparison is made the value of i + j increases by one unit. This value, observed at the time of comparing symbols, never decreases. Hence, there are at most n successful comparisons. If the first comparison is successful then we have no unsuccessful comparison for position i = 0. We conclude that: T(n) < n + n — m = 2n — m. For pat = ab and text = aaaa... Shift\j]); end; return(false) Knuth-Morris-Pratt algorithm We have not yet taken into account the full invariant invl of algorithm bruteforcel, but only its weaker version invl'.

Claim 2. The length of the overlap of the current match and the previous match is smaller than k. Proof, (of the claim) Recall that by a match we mean a scanned part of the text without the mismatch position. The period v is a suffix of the pattern. We already know, from Claim 1, that the end of the previous match cannot end at a critical position. Hence, if the overlap is at least k long, then v occurs inside the current match with the end position not placed at a critical position. The primitive word v then properly overlaps itself in a text in which the periodicity is v.